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NASA NTRS1971

Observations and reduction techniques of the US Coast and Geodetic Survey

The plate reduction method and other aspects of the world geodetic net are described. About one hundred stars are used as reference stars on the PAGEOS satellite plates obtained on 300mm f.1. and later 450mm f.1. Wild BC-4 cameras. The measurements are made manually now on diapositives since these can, according to experience, be measured with higher accuracy and are adjusted following a rigorous projective geometric model. The satellite images are analytically represented by a polynomial curve, of which seven fictitious points are regarded as the result from this plate. Some other aspects of the program are also discussed.

Schmid, H. H.↗

NASA NTRS1994

Separation of lithospheric, external, and core components of the south polar geomagnetic field at satellite altitudes

We present a new approach to producing scalar Magsat magnetic anomaly maps based on correlation coefficient filtering and the use of almost all of the available orbits. Our method differs from earlier techniques with respect to the following: (1) Passes are selected based on their variance properties rather than planetary indices such as Kp. (2) The core field model is least squares fit to individual passes and subsequently removed instead of substracting the model directly. This technique replaces band pass filtering or polynomial trend removal methods. (3) Each selected pass is sorted geographically and by local time, placed into one of four different altitude bands, and correlation coefficient filtered with the two adjacent passes. The filtering is the second step toward isolating the static lithospheric signal from the more dynamic external field signals. (4) Least squares collocation is used to grid the correlated passes; subsequently, the dawn and dusk maps are also correlation filtered providing another step toward removal of external fields. (5) The four resultant total field maps are continued to a common altitude and again correlation filtered for the static lithospheric anomalies. (6) The filtered results are then averaged together to provide a new total field map of the lithosphere south of 40 deg S latitude. Our total field map differs from previous efforts over the crustal blocks of West Antarctica. We obtained a positive anomaly over Edward VII Peninsula, extending into the Byrd subglacial basin and obtained a negative anomaly over the Ellsworth Mountains and parts of the Byrd subglacial basin. Also, a positive anomaly extending from the Ross Sea to offshore Wilkes Land is present in our map; however, this feature is absent in other maps. Positive anomalies marking the Weddell Sea in previous efforts are not present in our map. Prominent external field anomalies in the quadrant offshore of Wilkes Land are present in all previous efforts; however, these signatures are reduced in our total field map. This map significantly advances efforts for the identification and interpretation of magnetic anomalies of the south polar lithosphere.

Alsdorf, Douglas E.↗

NASA NTRS2015

Subharmonic Oscillations and Chaos in Dynamic Atomic Force Microscopy

The increasing use of dynamic atomic force microscopy (d-AFM) for nanoscale materials characterization calls for a deeper understanding of the cantilever dynamics influencing scan stability, predictability, and image quality. Model development is critical to such understanding. Renormalization of the equations governing d- AFM provides a simple interpretation of cantilever dynamics as a single spring and mass system with frequency dependent cantilever stiffness and damping parameters. The renormalized model is sufficiently robust to predict the experimentally observed splitting of the free-space cantilever resonance into multiple resonances upon cantilever-sample contact. Central to the model is the representation of the cantilever sample interaction force as a polynomial expansion with coefficients F(sub ij) (i,j = 0, 1, 2) that account for the effective interaction stiffness parameter, the cantilever-to-sample energy transfer, and the amplitude of cantilever oscillation. Application of the Melnikov method to the model equation is shown to predict a homoclinic bifurcation of the Smale horseshoe type leading to a cascade of period doublings with increasing drive displacement amplitude culminating in chaos and loss of image quality. The threshold value of the drive displacement amplitude necessary to initiate subharmonic generation depends on the acoustic drive frequency, the effective damping coefficient, and the nonlinearity of the cantilever-sample interaction force. For parameter values leading to displacement amplitudes below threshold for homoclinic bifurcation other bifurcation scenarios can occur, some of which lead to chaos.

Cantrell, John H.↗

NASA NTRS1992

Uses and limitations of the Tsyganenko magnetic field models

The Tsyganenko models contain 'modules' for two external current systems, the tail and ring currents; the former is the more detailed. No specific modules, however, cover the magnetopause and the Birkeland current system. The effects of these currents are represented by an all-purpose 'polynomial', and the resulting formulas involve about 30 parameters that specify the model. The dependence on magnetospheric indices is also simplified.

Stern, David P.↗

NASA NTRS2005

Uncertainty Modeling for Robustness Analysis of Control Upset Prevention and Recovery Systems

Formal robustness analysis of aircraft control upset prevention and recovery systems could play an important role in their validation and ultimate certification. Such systems (developed for failure detection, identification, and reconfiguration, as well as upset recovery) need to be evaluated over broad regions of the flight envelope and under extreme flight conditions, and should include various sources of uncertainty. However, formulation of linear fractional transformation (LFT) models for representing system uncertainty can be very difficult for complex parameter-dependent systems. This paper describes a preliminary LFT modeling software tool which uses a matrix-based computational approach that can be directly applied to parametric uncertainty problems involving multivariate matrix polynomial dependencies. Several examples are presented (including an F-16 at an extreme flight condition, a missile model, and a generic example with numerous crossproduct terms), and comparisons are given with other LFT modeling tools that are currently available. The LFT modeling method and preliminary software tool presented in this paper are shown to compare favorably with these methods.

Belcastro, Christine M.↗

NASA NTRS1976

Alternative methods for the design of jet engine control systems

Various alternatives to linear quadratic design methods for jet engine control systems are discussed. The main alternatives are classified into two broad categories: nonlinear global mathematical programming methods and linear local multivariable frequency domain methods. Specific studies within these categories include model reduction, the eigenvalue locus method, the inverse Nyquist method, polynomial design, dynamic programming, and conjugate gradient approaches.

Sain, M. K.↗

NASA NTRS1977

Compliance calibration of specimens used in the R-curve practice

The compliance calibrations for the compact (CS) and crack-line-wedge-loaded (CLWL) specimens have been determined by experimental measurements and by boundary-collocation analysis. The CS and CLWL specimen configurations were modeled more accurately than those used in previous analytical investigations. Polynomial expressions for the compliance at various stations along the crack line for CS and CLWL specimens are presented. The compliance calibrations for the center-crack tension (CCT) specimen have been determined theoretically by boundary-collocation and finite-element analysis. The calculated compliance values for the CCT specimen are compared with values obtained from the Irwin-Westergaard expression and from a modification to the Irwin-Westergaard expression proposed by Eftis and Liebowitz. The Eftis-Liebowitz expression was found to be in good agreement (plus or minus 2 percent) with both analyses for crack aspect ratios up to 0.8 and for gage half-span to specimen width ratios up to 0.5.

Mccabe, D. E.↗

NASA NTRS1979

Differences between seasonal and mean annual energy balance model calculations of climate and climate sensitivity

The paper extends a simple Budyko-Sellers mean annual energy balance climate model with diffusive transport to include a seasonal cycle. In the model the latitudinal distribution of the zonal average surface temperature is represented by Legendre polynomials and its time-dependence by a Fourier sine-cosine series, and it has three parameters adjusted so that the observed amplitudes of the Northern Hemisphere zonal mean surface temperature are recovered. The seasonal model is used to reveal how the annual mean climate and the sensitivity to changes in incident radiation differ from the predictions obtained with the corresponding mean annual model. The distribution of the incident solar radiation in the models is shown to be insensitive to changes in the eccentricity and the longitude of perihelion and sensitive only to changes in the obliquity of the earth, and for past orbital changes both the seasonal and the mean annual model fail to produce glacial advances of the magnitude that are thought to have occurred.

North, G. R.↗

NASA NTRS1984

Aberrations of varied line-space grazing incidence gratings in converging light beams

Analyses of the imaging properties of several designs for varied-line space gratings in converging beams of light in grazing-incidence spectrometers are presented. An explicit model is defined for the case of a plane-reflection grating intercepting light that converges and is reflected to a stigmatic point associated with the zero-order image of the grating. Smooth spatial variation of the grating constant then permits aberration correction. The aberrations are expressed as polynomials in the grating lens coordinates using power series expansions. Application of the model is illustrated in terms of aberrations experienced with the short wavelength spectrometer on the EUVE satellite. Attention is given to straight and parallel in-plane grooves, curved groove in-plane designs and off-plane grooves. Aberrations due to dispersions and misalignment are also considered.

Hettrick, M. C.↗

NASA NTRS1974

A Monte Carlo investigation of experimental data requirements for fitting polynomial functions

This report examines the extent to which sample size affects the accuracy of a low order polynomial approximation of an experimentally observed quantity and establishes a trend toward improvement in the accuracy of the approximation as a function of sample size. The task is made possible through a simulated analysis carried out by the Monte Carlo method, in which data are generated by using several transcendental or algebraic functions as models. Contaminated data of varying amounts are fitted to linear quadratic or cubic polynomials, and the behavior of the mean-squared error of the residual variance is determined as a function of sample size. Results indicate that the effect of the size of the sample is significant only for relatively small sample sizes and diminishes drastically for moderate and large amounts of experimental data.

Canavos, G. C.↗

NASA NTRS1974

An empirical analysis of the quantitative effect of data when fitting quadratic and cubic polynomials

A study is made of the extent to which the size of the sample affects the accuracy of a quadratic or a cubic polynomial approximation of an experimentally observed quantity, and the trend with regard to improvement in the accuracy of the approximation as a function of sample size is established. The task is made possible through a simulated analysis carried out by the Monte Carlo method in which data are simulated by using several transcendental or algebraic functions as models. Contaminated data of varying amounts are fitted to either quadratic or cubic polynomials, and the behavior of the mean-squared error of the residual variance is determined as a function of sample size. Results indicate that the effect of the size of the sample is significant only for relatively small sizes and diminishes drastically for moderate and large amounts of experimental data.

Canavos, G. C.↗

NASA NTRS1972

A feedback control model for network flow with multiple pure time delays

A control model describing a network flow hindered by multiple pure time (or transport) delays is formulated. Feedbacks connect each desired output with a single control sector situated at the origin. The dynamic formulation invokes the use of differential difference equations. This causes the characteristic equation of the model to consist of transcendental functions instead of a common algebraic polynomial. A general graphical criterion is developed to evaluate the stability of such a problem. A digital computer simulation confirms the validity of such criterion. An optimal decision making process with multiple delays is presented.

Press, J.↗

NASA NTRS1982

Crustal interpretation of the MAGSAT data in the continental United States

The processing of MAGSAT scalar data to construct a crustal magnetic anomaly map over the continental U.S. involves removal of the reference field model, a path-by-path subtraction of a low order polynomial through a least-squares fit to reduce orbital offset errors, and a two dimensional spectral filtering to mitigate the spectral bias induced by the path-by-path orbital correction scheme. The resultant anomaly map shows reasonably good correlations with an aeromagnetic map derived from the project MAGNET. Prominent satellite magnetic anomalies are identified in terms of geological provinces and age boundaries. An inversion method was applied to MAGSAT data which produces both the Curie depth topography and laterally varying magnetic susceptibility of the crust. A contoured Curie depth map thus derived shows general agreements with a crustal thickness map based on seismic data.

Won, I. J.↗

NASA NTRS1990

Analysis of spectral projectors in one-dimensional domains

A class of projection operators with values in a subspace of polynomials is analyzed. These projection operators are related to the Hilbert spaces involved in the numerical analysis of spectral methods. They are, in the first part of the paper, the standard Sobolev spaces and, in the second part, some weighted Sobolev spaces, the weight of which is related to the orthoronality relation satisfied by the Chebyshev polynomials. These results are used to study the approximation of a model fourth-order problem.

Maday, Y.↗

NASA NTRS2010

Focal Plane Geometry Characterization of the Kepler Mission

The Kepler Mission focal plane contains 42 charge-coupled device (CCD) photodetectors. Each CCD is composed of 2.2 million square pixels, 27 micrometers on a side, arranged in a grid of 2,200 columns by 1,044 rows. The science goals of the Kepler Mission require that the position of each CCD be determined with an accuracy of 0.1 pixels, corresponding to 2.7 micrometers or 0.4 seconds of arc, a level which is not achievable through pre-flight metrology. We describe a technique for determining the CCD positioning using images of the Kepler field of view (FOV) obtained in flight. The technique uses the fitted centroid row and column positions of 400 pre-selected stars on each CCD to obtain empirical polynomials which relate sky coordinates (right ascension and declination) to chip coordinates (row and column). The polynomials are in turn evaluated to produce constraints for a nonlinear model fit which directly determines the model parameters describing the location and orientation of each CCD. The focal plane geometry characterization algorithm is itself embedded in an iterative process which determines the focal plane geometry and the Pixel Response Function for each CCD in a self-consistent manner. In addition to the fully automated calculation, a person-in-the-loop implementation was developed to allow an initial determination of the geometry in the event of large misalignments, achieving a much looser capture tolerance for more modest accuracy and reduced automation.

ccd↗

NASA NTRS1989

Background stratospheric aerosol reference model

In this analysis, a reference background stratospheric aerosol optical model is developed based on the nearly global SAGE 1 satellite observations in the non-volcanic period from March 1979 to February 1980. Zonally averaged profiles of the 1.0 micron aerosol extinction for the tropics and the mid- and high-altitudes for both hemispheres are obtained and presented in graphical and tabulated form for the different seasons. In addition, analytic expressions for these seasonal global zonal means, as well as the yearly global mean, are determined according to a third order polynomial fit to the vertical profile data set. This proposed background stratospheric aerosol model can be useful in modeling studies of stratospheric aerosols and for simulations of atmospheric radiative transfer and radiance calculations in atmospheric remote sensing.

Mccormick, M. P.↗

NASA NTRS1998

Comparison of Response Surface and Kriging Models for Multidisciplinary Design Optimization

In this paper, we compare and contrast the use of second-order response surface models and kriging models for approximating non-random, deterministic computer analyses. After reviewing the response surface method for constructing polynomial approximations, kriging is presented as an alternative approximation method for the design and analysis of computer experiments. Both methods are applied to the multidisciplinary design of an aerospike nozzle which consists of a computational fluid dynamics model and a finite-element model. Error analysis of the response surface and kriging models is performed along with a graphical comparison of the approximations, and four optimization problems m formulated and solved using both sets of approximation models. The second-order response surface models and kriging models-using a constant underlying global model and a Gaussian correlation function-yield comparable results.

Simpson, Timothy W.↗

NASA NTRS1981

Global empirical models of ionospheric electron temperature in the upper F-region and plasmasphere based on in situ measurements from the Atmosphere Explorer-C, ISIS-1 and ISIS-2 satellites

Langmuir probe measurements of electron temperature, T sub e, in the vicinity of 300, 400, 1400 and 3000 km from the Atmosphere Explorer-C and the ISIS satellites have been employed to construct empirical models of the global distribution of T sub e at each of these altitudes. Legendre polynomials are employed to describe the observations at solstice and equinox in terms of dip latitude and local time. Sources of T sub e variations, such as solar activity, magnetic activity and longitude are found to be of second order importance, although they are resolvable in some cases by comparisons of the data with the model. The behavior of T sub e at the altitudes of these models is discussed in terms of its implications for our understanding of the energy exchange between the F-region and the plasmasphere.

Brace, L. H.↗